---
tag: 0NB8VPB93V
kind: definition
name: Composition series
msc: ["20E15", 20D30]
defines: 0XV2A49X6Q
uses: [096HN851YZ, 0C093D1T4M]
sources:
  - url: https://en.wikipedia.org/wiki/Composition_series
    locator: For groups
---

## Definition

Let $G$ be a [[group|G5S1M9]].
A composition series of $G$ is a [[subnormal series|0ZZR8D7Y1Z]] $\{e\} = G_0 \trianglelefteq G_1 \trianglelefteq \cdots \trianglelefteq G_n = G$ in which each factor $G_i / G_{i-1}$ is a [[simple group|0YMYMNQ0QZ]].

## Remarks

Equivalently, the inclusions are strict and each $G_{i-1}$ is a maximal proper normal subgroup of $G_i$.
Each finite group has a composition series, but the group $\mathbb{Z}$ has none.
For example, $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 3 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ and $\{0\} \trianglelefteq \langle 6 \rangle \trianglelefteq \langle 2 \rangle \trianglelefteq \mathbb{Z}/12\mathbb{Z}$ are composition series.

## Explanations

### L1 human

A composition series cuts a group into layers that cannot be cut further.
Each layer is a simple group.
It is like the factorisation of an integer into prime numbers.